Chapter 11 - SECTION 11.1 (PAGE 597) R. A. ADAMS: CALCULUS...

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SECTION 11.1 (PAGE 597) R. A. ADAMS: CALCULUS CHAPTER 11. VECTOR FUNCTIONS AND CURVES Section 11.1 Vector Functions of One Variable (page 597) 1. Position: r = i + t j Velocity: v = j Speed: v = 1 Acceleration : a = 0 Path: the line x = 1inthe xy -plane. 2. Position: r = t 2 i + k Velocity: v = 2 t i Speed: v = 2 | t | Acceleration : a = 2 i Path: the line z = 1, y = 0. 3. Position: r = t 2 j + t k Velocity: v = 2 t j + k Speed: v = 4 t 2 + 1 Acceleration : a = 2 j Path: the parabola y = z 2 in the plane x = 0. 4. Position: r = i + t j + t k Velocity: v = j + k Speed: v = 2 Acceleration : a = 0 Path: the straight line x = 1, y = z . 5. Position: r = t 2 i t 2 j + k Velocity: v = 2 t i 2 t j Speed: v = 2 2 t Acceleration: a = 2 i 2 j Path: the half-line x =− y 0, z = 1. 6. Position: r = t i + t 2 j + t 2 k Velocity: v = i + 2 t j + 2 t k Speed: v = 1 + 8 t 2 Acceleration: a = 2 j + 2 k Path: the parabola y = z = x 2 . 7. Position: r = a cos t i + a sin t j + ct k Velocity: v a sin t i + a cos t j + c k Speed: v = a 2 + c 2 Acceleration: a a cos t i a sin t j Path: a circular helix. 8. Position: r = a cos ω t i + b j + a sin ω t k Velocity: v a ω sin ω t i + a ω cos ω t k Speed: v =| a ω | Acceleration: a a ω 2 cos ω t i a ω 2 sin ω t k Path: the circle x 2 + z 2 = a 2 , y = b . 9. Position: r = 3 cos t i + 4 cos t j + 5 sin t k Velocity: v 3 sin t i 4 sin t j + 5 cos t k Speed: v = 9 sin 2 t + 16 sin 2 t + 25 cos 2 t = 5 Acceleration : a 3 cos t i 4 cos t j 5 sin t k r Path: the circle of intersection of the sphere x 2 + y 2 + z 2 = 25 and the plane 4 x = 3 y . 10. Position: r = 3 cos t i + 4 sin t j + t k Velocity: v 3 sin t i + 4 cos t j + k Speed: v = 9 sin 2 t + 16 cos 2 t + 1 = 10 + 7 cos 2 t Acceleration : a 3 cos t i 4 sin t j = t k r Path: a helix (spiral) wound around the elliptic cylinder ( x 2 / 9 ) + ( y 2 / 16 ) = 1. 11. Position: r = ae t i + be t j + ce t k Velocity and acceleration: v = a = r Speed: v = e t a 2 + b 2 + c 2 Path: the half-line x a = y b = z c > 0. 12. Position: r = at cos ω t i + sin ω t j + b ln t k Velocity: v = a ( cos ω t ω t sin ω t ) i + a ( sin ω t + ω t cos ω t ) j + ( b / t ) k Speed: v = ± a 2 ( 1 + ω 2 t 2 ) + ( b 2 / t 2 ) Acceleration: a a ω( 2 sin ω t + ω cos ω t ) i + a 2 cos ω t ω sin ω t ) j ( b / t 2 ) k Path: a spiral on the surface x 2 + y 2 = a 2 e z / b . 13. Position: r = e t cos ( e t ) i + e t sin ( e t ) j e t k Velocity: v ² e t cos ( e t ) + sin ( e t ) ³ i ² e t sin ( e t ) cos ( e t ) ³ j e t k Speed: v = 1 + e 2 t + e 2 t Acceleration: a = ² ( e t e t ) cos ( e t ) + sin ( e t ) ³ i + ² ( e t e t ) sin ( e t ) cos ( e t ) ³ j e t k Path: a spiral on the surface z ± x 2 + y 2 1. 14. Position: r = a cos t sin t i + a sin 2 t j + a cos t k = a 2 sin 2 t i + a 2 ² 1 cos 2 t ³ j + a cos t k Velocity: v = a cos 2 t i + a sin 2 t j a sin t k Speed: v = a 1 + sin 2 t Acceleration: a 2 a sin 2 t i + 2 a cos 2 t j a cos t k Path: the path lies on the sphere x 2 + y 2 + z 2 = a 2 ,on the surface defined in terms of spherical polar coordinates by φ = θ , on the circular cylinder x 2 + y 2 = ay , and on the parabolic cylinder + z 2 = a 2 . Any two of these surfaces serve to pin down the shape of the path.
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This note was uploaded on 12/16/2009 for the course FEW, FEWEB 400567 taught by Professor Moerdersen during the Fall '09 term at Vrije Universiteit Amsterdam.

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Chapter 11 - SECTION 11.1 (PAGE 597) R. A. ADAMS: CALCULUS...

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